Some excellent responses on cow vs. dog.  I'd add that part of the use of
dogs is protection -- and it always has been, even when dogs were wolves.  I
hike alone a lot and feel relatively safe with dog + Mace + martial arts
training.  Without dog, here in the U.S., I'd definitely want gun.  My dog
is also trained for search and rescue...admittedly, I've never tried that
training regimen with a cow, but I have my doubts it'd work, as cows, being
herbivores, don't use scent in the same way.  My dog also sits on me feet in
the winter and keeps them warm.  I've tried to get a cow to sit on my feet
but...well, maybe I'm just not using the right technique.

Thus, 'my' dog--that is, the dog whom I feed and who therefore has fealty to
the pack led by me--is useful to me for companionship and far more; a cow is
useful to me for steaks.

And I was a vegetarian for nine months and became increasingly weak and
mentally confused.  So I know all people cannot be healthy as vegetarians,
though I'm willing to admit that this only supports that all but
one--me--might be.